Theorem isdivrngo 34286

Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
isdivrngo	⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Proof of Theorem isdivrngo

Dummy variables 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4876	. . . . 5 ⊢ (𝐺DivRingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps)
2		df-drngo 34285	. . . . . . 7 ⊢ DivRingOps = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ RingOps ∧ (𝑦 ↾ ((ran 𝑥 ∖ {(GId‘𝑥)}) × (ran 𝑥 ∖ {(GId‘𝑥)}))) ∈ GrpOp)}
3	2	relopabi 5482	. . . . . 6 ⊢ Rel DivRingOps
4	3	brrelex1i 5397	. . . . 5 ⊢ (𝐺DivRingOps𝐻 → 𝐺 ∈ V)
5	1, 4	sylbir 227	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps → 𝐺 ∈ V)
6	5	anim1i 608	. . 3 ⊢ ((⟨𝐺, 𝐻⟩ ∈ DivRingOps ∧ 𝐻 ∈ 𝐴) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
7	6	ancoms 452	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ ⟨𝐺, 𝐻⟩ ∈ DivRingOps) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
8		rngoablo2 34245	. . . . 5 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)
9		elex 3429	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
10	8, 9	syl 17	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
11	10	ad2antrl 719	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐺 ∈ V)
12		simpl 476	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐻 ∈ 𝐴)
13	11, 12	jca 507	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
14		df-drngo 34285	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
15	14	eleq2i 2898	. . 3 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)})
16		opeq1 4625	. . . . . 6 ⊢ (𝑔 = 𝐺 → ⟨𝑔, ℎ⟩ = ⟨𝐺, ℎ⟩)
17	16	eleq1d 2891	. . . . 5 ⊢ (𝑔 = 𝐺 → (⟨𝑔, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, ℎ⟩ ∈ RingOps))
18		rneq 5587	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
19		fveq2 6437	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
20	19	sneqd 4411	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(GId‘𝑔)} = {(GId‘𝐺)})
21	18, 20	difeq12d 3958	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (ran 𝑔 ∖ {(GId‘𝑔)}) = (ran 𝐺 ∖ {(GId‘𝐺)}))
22	21	sqxpeqd 5378	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
23	22	reseq2d 5633	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
24	23	eleq1d 2891	. . . . 5 ⊢ (𝑔 = 𝐺 → ((ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
25	17, 24	anbi12d 624	. . . 4 ⊢ (𝑔 = 𝐺 → ((⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
26		opeq2 4626	. . . . . 6 ⊢ (ℎ = 𝐻 → ⟨𝐺, ℎ⟩ = ⟨𝐺, 𝐻⟩)
27	26	eleq1d 2891	. . . . 5 ⊢ (ℎ = 𝐻 → (⟨𝐺, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
28		reseq1 5627	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
29	28	eleq1d 2891	. . . . 5 ⊢ (ℎ = 𝐻 → ((ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
30	27, 29	anbi12d 624	. . . 4 ⊢ (ℎ = 𝐻 → ((⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
31	25, 30	opelopabg 5221	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
32	15, 31	syl5bb 275	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
33	7, 13, 32	pm5.21nd 836	1 ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ∖ cdif 3795  {csn 4399  ⟨cop 4405   class class class wbr 4875  {copab 4937   × cxp 5344  ran crn 5347   ↾ cres 5348  ‘cfv 6127  GrpOpcgr 27895  GIdcgi 27896  AbelOpcablo 27950  RingOpscrngo 34230  DivRingOpscdrng 34284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-1st 7433  df-2nd 7434  df-rngo 34231  df-drngo 34285

This theorem is referenced by:  zrdivrng  34289  isdrngo1  34292